Factor the following expression: $5$ $x^2$ $-29$ $x+$ $20$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(20)} &=& 100 \\ {a} + {b} &=& & & {-29} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $100$ and add them together. The factors that add up to ${-29}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-4}$ and ${b}$ is ${-25}$ $ \begin{eqnarray} {ab} &=& ({-4})({-25}) &=& 100 \\ {a} + {b} &=& {-4} + {-25} &=& -29 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 {-4}x {-25}x +{20} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 {-4}x) + ({-25}x +{20}) $ Factor out the common factors: $ x(5x - 4) - 5(5x - 4) $ Notice how $(5x - 4)$ has become a common factor. Factor this out to find the answer. $(5x - 4)(x - 5)$